Recent papers, [1],[2] & [3], describe some algorithms for linear first kind integral equations. These algorithms are based on augmented Galerkin method and Cross-validation scheme [5]. The results show that, these algorithms work well for linear equations. In this paper we apply algorithms of [1] & [2] on non-linear first kind integral equations of Hammerstein type with bounded solution. In order to obtain a posteriori error estimate, we apply fifteen-point Gauss-Kronrod quadrature rule [4]. Finally, we give a number of numerical examples showing that the algorithms work well in practice.